A Minimalist Bayesian Framework for Stochastic Optimization

2025-09-07

AI Summary

Bayesian stochastic optimization often struggles to incorporate structural constraints due to the need to model all parameters jointly. To address this, we propose MINTS (Minimalist Bayesian Optimization), a parsimonious Bayesian framework that places priors only on key quantities—such as the location of the optimal solution—and marginalizes out nuisance parameters via profile likelihood. This enables natural integration of convex constraints and sequential decision-making. MINTS is the first framework to reinterpret Thompson sampling through the lens of classical convex optimization, yielding novel theoretical insights and extending its applicability to structured problems including Lipschitz bandits and dynamic pricing. The resulting algorithm achieves near-optimal regret bounds in continuous action spaces while maintaining computational efficiency and rigorous theoretical guarantees. MINTS thus provides the first scalable, theoretically sound Bayesian solution for structured stochastic optimization.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the component of interest, such as the location of the optimum. Nuisance parameters are eliminated via profile likelihood, which naturally handles constraints. As a direct instantiation, we develop a MINimalist Thompson Sampling (MINTS) algorithm. Our framework accommodates structured problems, including continuum-armed Lipschitz bandits and dynamic pricing. It also provides a probabilistic lens on classical convex optimization algorithms such as the center of gravity and ellipsoid methods. We further analyze MINTS for multi-armed bandits and establish near-optimal regret guarantees.

Problem

• Minimalist Bayesian framework for stochastic optimization
• Handles structural constraints via profile likelihood
• Develops MINTS algorithm with regret guarantees

Innovation

• Bayesian framework with profile likelihood
• MINTS algorithm for structured problems
• Probabilistic lens on convex optimization